Navigation systems are one example of nonlinear dynamic systems. One of the problems associated with navigation system development is the estimation of the various states of the dynamic system. Such estimations typically utilize navigation system software. The extended Kalman filter (EKF) has been used within navigation system software to make such estimations. In some navigation systems, the EKF applies a Taylor series expansion for nonlinear system and observation equations, and utilizes first order terms to apply the well-known linear Kalman filter theory, where the probability density function (PDF) is assumed to be Gaussian.
In practice, however, the EKF has shown several limitations. One of the limitations is that only small errors are allowed to be input into the EKF. Otherwise, in the presence of nonlinear error behavior, the first-order approximations can cause biased solutions and inconsistencies in updating of the covariance matrix, which can lead to filter instability. While second-order versions of the EKF exist, their increased implementation and computation complexity tend to prohibit their use.
A popular technique to improve the first-order approach is the iterated EKF, which effectively iterates the EKF equations at the current time observation by redefining the nominal state estimate and re-linearizing the measurement equations. The iterated EKF is capable of providing better performance than the basic EKF, especially in the case of significant nonlinearity in the measurement function.
Recently, low-cost MEMS-based sensors have become available and affordable for utilization in inertial navigation systems (INS). Such INS applications include airplane navigation, determination of position, and guidance. Most navigation systems contain a GPS range measuring device and an INS which provides data relating to an angular velocity, velocity, and azimuth measuring, which are used in combination to measure motion of a mobile object (e.g., airplane). The navigation systems also contain range error estimating devices. Based on the output of the error estimating devices, position of the mobile object can be determined. The error estimating devices are sometimes implemented using Kalman filters and averaging processes.
The outputs of the various measuring devices are thereby corrected using Kalman filters and the like so that the position of the mobile object can be estimated with a relatively high level of accuracy without using a high precision sensor. However, due to the nature of high noise, nonlinear effects, and imprecise measurements associated with low cost MEMS sensors, traditional EKF estimation will degrade with time and become unreliable. Therefore, the accuracy of the INS is limited, especially when GPS data is not available. Most nonlinear Kalman filters can be used to improve an estimation error, however, such implementations are difficult. More specifically, such implementations are difficult to tune and additionally it is difficult to switch estimation schemes because the nonlinear effect only shows up in certain scenarios.
Current navigation systems that use an inexpensive inertial measurement unit (IMU), such as an IMU based on MEMS devices, also exhibit difficulties in gyrocompass alignment due to a large “turn on” bias and other related errors.